Bone rotation behaviour based on hierarchy of individual bone axis like Maya
Set the gizmo to manipulate all the selected items individually:



If you want the gizmo to act on each bone's local axis, rather than global, set it here:



It will take bones roll in mind.

This is however not a very convenient way to rig fingers. The easiest is to use Rigify and create finger rigs with it:

I need to add finger rig to a model that is already rigged

The idea is to have extra controls for rough pose and also access to individual bones for fine tuning. 

It can be driven through constraints or with drivers, it can behave like this: